WO2019204452A1 - Systèmes et procédés de caractérisation de matériaux poroélastiques - Google Patents

Systèmes et procédés de caractérisation de matériaux poroélastiques Download PDF

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Publication number
WO2019204452A1
WO2019204452A1 PCT/US2019/027886 US2019027886W WO2019204452A1 WO 2019204452 A1 WO2019204452 A1 WO 2019204452A1 US 2019027886 W US2019027886 W US 2019027886W WO 2019204452 A1 WO2019204452 A1 WO 2019204452A1
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WIPO (PCT)
Prior art keywords
time
indentation
data
poroelastic
force
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PCT/US2019/027886
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English (en)
Inventor
Haiying Huang
Ming Liu
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Georgia Tech Research Corporation
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Priority to US17/048,670 priority Critical patent/US11747251B2/en
Publication of WO2019204452A1 publication Critical patent/WO2019204452A1/fr

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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N3/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N3/40—Investigating hardness or rebound hardness
    • G01N3/42—Investigating hardness or rebound hardness by performing impressions under a steady load by indentors, e.g. sphere, pyramid
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N3/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N3/02—Details

Definitions

  • the present disclosure relates generally to characterization of solid materials. Particularly, embodiments of the present disclosure relate to systems and methods for characterizing poroelastic materials.
  • the present invention relates to systems and methods for characterizing poroelastic materials.
  • An exemplary embodiment of the present invention can provide a method for characterizing poroelastic materials, comprising: obtaining experimental data for a material, comprising at least: time data and indentation force data; indicating a first asymptote of the indentation force data at a corresponding first time from the time data; indicating a second asymptote of the indentation force data at a corresponding second time from the time data; selecting a corresponding master curve from a plurality of theoretical master curves based on the ratio of the first and the second asymptotes; and calculating a value for a property of the material by matching the experimental data with the corresponding master curve.
  • the material can be a poroelastic solid.
  • the obtaining the experimental data can comprise: indenting, with an indentation tool, the poroelastic solid to a predetermined indentation depth; measuring a force required to maintain the indentation tool at the indentation depth to obtain the indentation force data; and recording the indentation force data with respect to the corresponding time data.
  • the indentation tool can comprise a rigid smooth sphere.
  • the rigid sphere can be selected from the group consisting of: permeable indenters and impermeable indenters.
  • the drainage condition on the surface of the poroelastic solid can be selected from the group consisting of: a permeable indenter on a fully permeable surface, a permeable/impermeable indenter on a fully impermeable surface, and an impermeable indenter on a fully permeable surface.
  • the first time from the time data can be an initial time, wherein the indentation force reaches a maximum.
  • the second time from the time data can be an ending time.
  • the desired material property can be the coefficient of hydraulic diffusion.
  • the experimental data can comprise at least a force relaxation behavior.
  • the determining a corresponding master curve can comprise: calculating a constant comprising the ratio of the first and the second asymptotes; and selecting a corresponding master curve from a theoretical solution corresponding to the value of the constant and the drainage condition of the contact surface.
  • the calculating the value of the property of the material can comprise fitting, using a fitting function predetermined by the full poroelastic solution, the experimental data to the corresponding master curve to obtain the value of the property of the material.
  • the plurality of master curves can be obtained by: establishing one or more governing equations; establishing one or more boundary conditions using Heaviside step displacement loading; transforming the one or more governing equations using Hankel transform in the Laplace domain to find the expressions for the two displacement functions in terms of integrals with three unknowns; transforming the one or more boundary conditions to the Laplace domain and matching the boundary conditions with the field quantities expressed through the displacement functions to obtain three equations for the three unknowns, which includes one or more Fredholm integral equations of the second kind; providing alternate integral expressions for the integral kernels in the Fredholm integral equations using one or more modified Struve functions; solving the Fredholm integral equations using a method of successive substitution; calculating a time domain solution by numerically inverting the solved Laplace domain equation; and integrating, based on a plurality of parameters of a specific poroelastic material, the time domain solution to obtain a plurality of master curves representing force relaxation.
  • the one or more governing equations can be obtained from one or more McNamee-Gibson displacement function methods.
  • the one or more governing equations can comprise one or more material constants.
  • the expressions of one or more field quantities can comprise the displacement function solutions from the one or more governing equations.
  • Another embodiment of the present disclosure can provide a system for characterizing poroelastic materials, comprising: an indentation tool, comprising a rigid smooth sphere; one or more force sensors; one or more processors; and at least one memory storing instructions that when executed by the one or more processors, cause the system to: indent, with the indentation tool, a material to a predetermined indentation depth; measure, using the one or more force sensors, a force required to maintain the indentation tool at the indentation depth to obtain indentation force data; record experimental data for the material, comprising at least: indentation force data and time data corresponding to the indentation force data; indicate a first asymptote of the indentation force data at a corresponding first time from the time data; indicate a second asymptote of the indentation force data at
  • the material can be a poroelastic solid.
  • the rigid sphere can be selected from the group consisting of: permeable indenters and impermeable indenters.
  • the drainage condition on the surface of the poroelastic solid can be selected from the group consisting of: a permeable indenter on a fully permeable surface, a permeable/impermeable indenter on a fully impermeable surface, and an impermeable indenter on a fully permeable surface.
  • the first time from the time data can be an initial time, wherein the indentation force reaches a maximum.
  • the second time from the time data can be an ending time.
  • the desired material property can be the coefficient of hydraulic diffusion.
  • the experimental data can comprise at least a force relaxation behavior.
  • the determining a corresponding master curve can comprise: calculating a constant comprising the ratio of the first and the second asymptotes; and selecting a corresponding master curve from a theoretical solution corresponding to the value of the constant and the drainage condition of the contact surface.
  • the calculating the value of the property of the material can comprise fitting, using a fitting function predetermined by the full poroelastic solution, the experimental data to the corresponding master curve to obtain the value of the property of the material.
  • the plurality of master curves can be obtained by storing instructions that when executed by the one or more processors, cause the system to: establish one or more governing equations; establish one or more boundary conditions using Heaviside step displacement loading; transform the one or more governing equations using Hankel transform in the Laplace domain to find the expressions for the two displacement functions in terms of integrals with three unknowns; transform the one or more boundary conditions to the Laplace domain and matching the boundary conditions with the field quantities expressed through the displacement functions to obtain three equations for the three unknowns, which includes one or more Fredholm integral equations of the second kind; provide alternate integral expressions for the integral kernels in the Fredholm integral equations using one or more modified Struve functions; solve the Fredholm integral equations using a method of successive substitution; calculate a time domain solution by numerically inverting the solved Laplace domain equation; and integrate, based on a plurality of parameters of a specific poroelastic material, the time domain solution to obtain a plurality of
  • the one or more governing equations can be obtained from one or more McNamee-Gibson displacement function methods.
  • the one or more governing equations can comprise one or more material constants.
  • the expressions of one or more field quantities can comprise the displacement function solutions from the one or more governing equations.
  • FIG. 1 is a flowchart of an exemplary embodiment of a method for characterizing poroelastic materials
  • FIG. 2 is a flowchart of an exemplary embodiment of a method for characterizing poroelastic materials
  • Fig. 3 is a graph of contact pressure at various times for an exemplary embodiment of a method for characterizing poroelastic materials
  • Fig. 4 is a graph of radial stress at dimensionless times for an exemplary embodiment of a method for characterizing poroelastic materials
  • Fig. 5 is a graph of distribution of pore pressure at dimensionless times for an exemplary embodiment of a method for characterizing poroelastic materials
  • Fig. 6 is a graph of a plurality of master curves for relaxation of normalized indentation force obtained from an exemplary embodiment of a method for characterizing poroelastic materials
  • FIG. 7 is a flowchart of an exemplary embodiment of a method for characterizing poroelastic materials.
  • Fig. 8 is a graph of a comparison between direct numerical integration of an oscillatory kernel and the alternative expression using an exemplary embodiment of the present disclosure.
  • Indentation of a poroelastic solid by a spherical -tip tool can be analyzed within the framework of Biot’s theory.
  • Embodiments of the present disclosure seek the response of the indentation force as well as the field variables as functions of time when the rigid indenter is loaded instantaneously to a fixed depth.
  • Three particular cases can be considered when the drainage condition of the surface of the semi -infinite domain is one of the following: 1) a permeable indenter on a fully permeable surface (case I or drained case), 2) a permeable/impermeable indenter on a fully impermeable surface (case II or undrained case), 3) an impermeable indenter on a fully permeable surface (case III or mixed case). Compressibility of both the fluid and solid phases can be taken into account.
  • a 2 and B t are functions of x and s to be determined through the boundary conditions.
  • the overbar is used here to denote the functions in the Laplace domain.
  • Heaviside step displacement loading can be applied to the spherical-tip indenter. Conformity is assumed at the frictionless contact surface. It can be shown that if a sphere is pressed to a fixed depth in a poroelastic medium, the contact radius in fact changes with time. It is unclear whether such a problem with a free and moving boundary can be solved analytically.
  • JC (t) is the Heaviside step function.
  • Constant w can be expressed explicitly using other material constants
  • J _ (cx) is the Bessel function of the first kind of order—1/2 and 6(s, m) satisfies a
  • Eq. 12 can then be evaluated numerically, where ⁇ (s * , x * ) is the unknown to be determined.
  • ⁇ (s * , x * ) is the unknown to be determined.
  • a different approach over known methods can be adopted by providing an alternative integral expression for N(s , x * , m * ).
  • the expression for iV(s * , x * , m * ) can be rewritten using one of the integral representations of the modified Struve functions, in which the oscillatory nature can be removed.
  • L ⁇ x is the Laplace inversion operator.
  • the ratio of the contact pressure is the same as the ratio of the indentation force between the small and large times, namely,
  • Indentation force at an intermediate time can be expressed in a normalized form
  • Eq. 17 shows that the normalized indentation force is a function of constant w only.
  • the two force asymptotes determine the material constants G/f, ⁇ (2h— 1)/h and the ratio of the two force asymptotes gives constant w.
  • the force relaxation curves in Fig. 6 can serve as the master curves to determine the diffusion coefficient c.
  • the force- relaxation curve appears to be insensitive to w. This means that the force-relaxation curve could be a rather reliable mean for determining the diffusivity coefficient c since the uncertainty in w does not have a strong effect on the force-relaxation behavior.
  • Eq. 17 can also be used to calculate the normalized indentation force for Case II. Replacing 6 t (s * , x * ) with 0 la (s t , x * ) gives the normalized force relaxation expression for Case III. Summary of the normalized indentation force relaxation for all three cases is shown in Fig. 6
  • FIGs. 1-2 and 10 illustrate exemplary embodiments of the presently disclosed systems and methods for characterizing poroelastic materials.
  • experimental data can be obtained from a material test, comprising at least time data and indentation force data.
  • the experimental data can be obtained using an indentation tool configured to indent a material to a predetermined depth.
  • the indentation tool can comprise a spherical indenter in the form of a rigid sphere.
  • the indenter can be a rigid smooth sphere.
  • the state of the indenter can be permeable or impermeable.
  • the experimental data can be obtained through one or more force sensors (e.g., two or more, three or more, four or more, or five or more).
  • a force sensor can be housed in the indentation tool to measure the force applied to the material.
  • Additional force sensors can be attached to the system in any configuration such that the applied force to the material can be obtained from the sensors.
  • the force sensors can be connected to one or more storage devices (e.g., two or more, three or more, four or more, or five or more).
  • Suitable examples of a storage device can include, but are not limited to, hard drives, hard disks, solid-state drives, removable universal serial bus (USB) drives, floppy disks, compact disks (CDs) and the like.
  • the one or more storage devices can be configured to store the experimental data, along with other data. It is understood that the storage devices can store more than the experimental data.
  • the one or more force sensors, the indentation tool, and/or the one or more storage devices can be connected to one or more processors (e.g., two or more, three or more, four or more, or five or more).
  • the one or more processors can be configured to execute instructions given to the system.
  • the one or more processors can be configured to cause the indentation tool to indent the material and can cause the one or more storage devices to begin recording the experimental data received from the one or more force sensors.
  • the system can comprise at least one memory configured to store instructions.
  • the instructions stored on the at least one memory can be executed by the one or more processors.
  • the presently disclosed method steps can be stored in the memory and executed by the one or more processors.
  • a first asymptote of the indentation force data can be indicated with a corresponding first time from the time data.
  • the first time from the time data can be the initial time and correspond with an initial indentation force asymptote.
  • the first asymptote of the indentation force data would be the initial force required to indent a material to a predetermined depth.
  • the indication can be received from the one or more processors. Additionally, the indication can be stored on the one or more storage devices along with the relevant portions of the experimental data.
  • a second asymptote of the indentation force data can be indicated with a corresponding second time from the time data.
  • the second time from the time data can be the ending time, or termination time of the test.
  • the second time can be the time tending towards infinity and corresponding with a steady-state indentation force asymptote.
  • This steady-state force value can be taken as the second asymptote.
  • the indication can be received from the one or more processors. Additionally, the indication can be stored on the one or more storage devices along with the relevant portions of the experimental data. [0102]
  • the ratio between the first and the second asymptotes can be calculated and used to select a corresponding master curve from a plurality of master curves. In some embodiments, the ratio between the first and the second asymptotes can be calculated as a constant, where each master curve from the plurality of master curves corresponds to a certain value of the constant. As mentioned above, the ratio of the asymptotes can be the ratio between the initial force required to maintain the indentation depth and the steady-state force required to maintain the indentation depth. In some embodiments, the plurality of master curves can be stored in the one or more storage devices. The calculation of the ratio can be performed by the one or more processors, which can then retrieve the correct corresponding master curve from the one or more storage devices.
  • the selected master curve can be matched with the experimental data to obtain desired material properties.
  • the data matching can be performed by the one or more processors.
  • the desired material properties can be calculated for the performed material test.
  • the desired material property can be the coefficient of hydraulic diffusion, hydraulic diffusivity (or coefficient of consolidation for soils), the hardness, the material toughness, and the like.
  • the calculating can be performed by the one or more processors. Further, the calculated material property values can be stored in the one or more storage devices for later use.
  • a method 200 for constructing master curves for a poroelastic material is disclosed herein.
  • the master curve construction can be carried out by the one or more processors.
  • the one or more processors can receive data input by a user based on the material being tested, such as material properties.
  • the one or more processors can store the master curves in one or more storage devices after construction.
  • one or more governing equations and one or more boundary conditions can be established.
  • the boundary conditions and the governing equations can be transformed into the Laplace domain.
  • alternate integral expressions can be provided for the integral kernels using one or more modified Struve functions.
  • the integral equations can be solved using a method of successive substitution.
  • a time domain solution can be calculated by numerically inverting the solved integral equations from the Laplace domain.
  • the time domain solutions can be integrated to obtain the plurality of master curves.
  • the master curves can be normalized to make time dimensionless, and/or to normalize the force between 0 and 1. Suitable examples of calculated master curves can be seen in Fig. 6.
  • a method 700 for characterizing poroelastic materials is disclosed herein.
  • the indentation force can be recorded as a function of time.
  • the indentation force can form a force relaxation curve.
  • the early and late time asymptotes can be denoted. The ratio between the asymptotes can be calculated to obtain a constant.
  • a corresponding master curve can be chosen to model the behavior of the material.
  • the indentation force can be normalized to be within 0 and 1, inclusive.
  • the experimental data can be matched with the corresponding master curve through data regression to obtain the desired material properties.

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Abstract

L'invention concerne des systèmes et des procédés de caractérisation de matériaux poroélastiques. L'indentation d'un solide porogène par un outil à pointe sphérique est analysée dans le cadre de la théorie de Biot. La présente invention fournit la réponse de la force d'indentation ainsi que les variables de champ en tant que fonctions temporelles lorsque le pénétrateur rigide est chargé instantanément à une profondeur fixe. Certains modes de réalisation de la présente invention prennent en considération le cas particulier de la surface du domaine semi-infini perméable et dans une condition drainée. La compressibilité du fluide et de phases solides est prise en compte. La procédure de solution fondée sur le procédé de fonction de déplacement McNamee-Gibson est adoptée.
PCT/US2019/027886 2018-04-17 2019-04-17 Systèmes et procédés de caractérisation de matériaux poroélastiques WO2019204452A1 (fr)

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